The temperature dependence of the conductivity of an intrinsic semiconductor is given by = ₀ e^ - E/2kT , where ₀ is a constant. Determine the temperature at which the conductivity of an intrinsic germanium semiconductor becomes twice its value at T = 300 K . Assume the energy gap for germanium is 0.650 eV and it stays constant as temperature increases.

Step-by-step solution

The conductivity at temperature T is given by = ₀ e^ - E/2kT . Let the initial temperature be T₁ = 300 K and the final temperature be T₂ . Given that ₂ = 2 ₁ , we have: ₀ e^ - E/2kT₂ = 2 ₀ e^ - E/2kT₁ Taking the natural logarithm on both sides: - E 2kT₂ = 2 - E 2kT₁ 1 T₂ = 1 T₁ - 2k 2 E Substituting the given values T₁ = 300 K , k = 8.625 10⁻⁵ eV/K , E = 0.650 eV , and 2 = 0.693 : 1 T₂ = 1 300 - 2 8.625 10⁻⁵ 0.693 0.650 1 T₂ = 3.333 10⁻³ - 1.839 10⁻⁴ 1 T₂ = 3.149 10⁻³ K ⁻¹ T₂ = 1 3.149 10⁻³ 317.5 K Rounding to the